Discoveries of marked similarities to high-Tc cuprate superconductors point to the realization of superconductivity in the doped J(eff) = 1/2 Mott insulator Sr2IrO4. Contrary to the mother compound of cuprate superconductors, several stacking patterns of in-plane canted antiferromagnetic moments have been reported, which are distinguished by the ferromagnetic components as -++-, ++++ and -+-+. In this paper, we clarify unconventional features of the superconductivity coexisting with -++- and -+-+ structures. Combining the group theoretical analysis and numerical calculations for an effective J(eff) = 1/2 model, we show unusual superconducting gap structures in the -++- state protected by nonsymmorphic magnetic space group symmetry. Furthermore, our calculation shows that the FuldeFerrell- Larkin-Ovchinnikov superconductivity is inevitably stabilized in the -+ -+ state since the oddparity magnetic -+-+ order makes the band structure asymmetric by cooperating with spin-orbit coupling. These unusual superconducting properties are signatures of magnetic multipole order in nonsymmorphic crystal.